where \(U^{\ell }, V^{\ell }, A^{\ell }, B^{\ell }, C^{\ell } \in \mathbb {R}^{d \times d}\)  are learnable parameters, Norm denotes the normalization layer (BatchNorm [1]}, LayerNorm [2]}), Aggr represents the neighborhood aggregation function (Sum, Mean or Max), \(\sigma \)  is the sigmoid function, and \(\odot \)  is the Hadamard product.
As inputs \(h_i^{\ell =0}\)  and \(e_{ij}^{\ell =0}\) , we use \(d\) -dimensional linear projections of the node coordinate \(x_i\)  and the euclidean distance \(\Vert x_i - x_j\Vert _2\) , respectively.
Anisotropic GNNs [3]}, [4]} have been shown to outperform simpler isotropic variants [5]}, [6]} across several challenging domains, including TSP [7]}.
